دانلود کتاب The Story of Proof: Logic and the History of Mathematics

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Cover
Contents
Preface
1. Before Euclid
1.1 The Pythagorean Theorem
1.2 Pythagorean Triples
1.3 Irrationality
1.4 From Irrationals to Infinity
1.5 Fear of Infinity
1.6 Eudoxus
1.7 Remarks
2. Euclid
2.1 Definition, Theorem, and Proof
2.2 The Isosceles Triangle Theorem and SAS
2.3 Variants of the Parallel Axiom
2.4 The Pythagorean Theorem
2.5 Glimpses of Algebra
2.6 Number Theory and Induction
2.7 Geometric Series
2.8 Remarks
3. After Euclid
3.1 Incidence
3.2 Order
3.3 Congruence
3.4 Completeness
3.5 The Euclidean Plane
3.6 The Triangle Inequality
3.7 Projective Geometry
3.8 The Pappus and Desargues Theorems
3.9 Remarks
4. Algebra
4.1 Quadratic Equations
4.2 Cubic Equations
4.3 Algebra as “Universal Arithmetick”
4.4 Polynomials and Symmetric Functions
4.5 Modern Algebra: Groups
4.6 Modern Algebra: Fields and Rings
4.7 Linear Algebra
4.8 Modern Algebra: Vector Spaces
4.9 Remarks
5. Algebraic Geometry
5.1 Conic Sections
5.2 Fermat and Descartes
5.3 Algebraic Curves
5.4 Cubic Curves
5.5 Bézout’s Theorem
5.6 Linear Algebra and Geometry
5.7 Remarks
6. Calculus
6.1 From Leonardo to Harriot
6.2 Infinite Sums
6.3 Newton’s Binomial Series
6.4 Euler’s Solution of the Basel Problem
6.5 Rates of Change
6.6 Area and Volume
6.7 Infinitesimal Algebra and Geometry
6.8 The Calculus of Series
6.9 Algebraic Functions and Their Integrals
6.10 Remarks
7. Number Theory
7.1 Elementary Number Theory
7.2 Pythagorean Triples
7.3 Fermat’s Last Theorem
7.4 Geometry and Calculus in Number Theory
7.5 Gaussian Integers
7.6 Algebraic Number Theory
7.7 Algebraic Number Fields
7.8 Rings and Ideals
7.9 Divisibility and Prime Ideals
7.10 Remarks
8. The Fundamental Theorem of Algebra
8.1 The Theorem before Its Proof
8.2 Early “Proofs” of FTA and Their Gaps
8.3 Continuity and the Real Numbers
8.4 Dedekind’s Definition of Real Numbers
8.5 The Algebraist’s Fundamental Theorem
8.6 Remarks
9. Non-Euclidean Geometry
9.1 The Parallel Axiom
9.2 Spherical Geometry
9.3 A Planar Model of Spherical Geometry
9.4 Differential Geometry
9.5 Geometry of Constant Curvature
9.6 Beltrami’s Models of Hyperbolic Geometry
9.7 Geometry of Complex Numbers
9.8 Remarks
10. Topology
10.1 Graphs
10.2 The Euler Polyhedron Formula
10.3 Euler Characteristic and Genus
10.4 Algebraic Curves as Surfaces
10.5 Topology of Surfaces
10.6 Curve Singularities and Knots
10.7 Reidemeister Moves
10.8 Simple Knot Invariants
10.9 Remarks
11. Arithmetization
11.1 The Completeness of R
11.2 The Line, the Plane, and Space
11.3 Continuous Functions
11.4 Defining “Function” and “Integral”
11.5 Continuity and Differentiability
11.6 Uniformity
11.7 Compactness
11.8 Encoding Continuous Functions
11.9 Remarks
12. Set Theory
12.1 A Very Brief History of Infinity
12.2 Equinumerous Sets
12.3 Sets Equinumerous with R
12.4 Ordinal Numbers
12.5 Realizing Ordinals by Sets
12.6 Ordering Sets by Rank
12.7 Inaccessibility
12.8 Paradoxes of the Infinite
12.9 Remarks
13. Axioms for Numbers, Geometry, and Sets
13.1 Peano Arithmetic
13.2 Geometry Axioms
13.3 Axioms for Real Numbers
13.4 Axioms for Set Theory
13.5 Remarks
14. The Axiom of Choice
14.1 AC and Infinity
14.2 AC and Graph Theory
14.3 AC and Analysis
14.4 AC and Measure Theory
14.5 AC and Set Theory
14.6 AC and Algebra
14.7 Weaker Axioms of Choice
14.8 Remarks
15. Logic and Computation
15.1 Propositional Logic
15.2 Axioms for Propositional Logic
15.3 Predicate Logic
15.4 Gödel’s Completeness Theorem
15.5 Reducing Logic to Computation
15.6 Computably Enumerable Sets
15.7 Turing Machines
15.8 TheWord Problem for Semigroups
15.9 Remarks
16. Incompleteness
16.1 From Unsolvability to Unprovability
16.2 The Arithmetization of Syntax
16.3 Gentzen’s Consistency Proof for PA
16.4 Hidden Occurrences of ε0 in Arithmetic
16.5 Constructivity
16.6 Arithmetic Comprehension
16.7 TheWeak Kőnig Lemma
16.8 The Big Five
16.9 Remarks
Bibliography
Index